The present invention relates to signal data processing and in particular to efficient reconstruction of non-uniformly sampled data for medical imaging.
The Fourier Transform (FT) is one of the most important signal processing techniques in use today. In particular, it finds a number of uses in medical imaging, including reconstruction of MRI images and Fourier reconstruction of CT images. In MRI applications, the FT is used to convert the acquired data into an image. The quality and accuracy of the image is of utmost importance in medical examinations.
The Fast Fourier Transform (FFT) is an efficient implementation of the FT, which is designed for data that is uniformly sampled in a grid having sides which are typically powers of two. In addition to the FFT, there are other signal and image processing techniques that require the input data to be sampled to a specific grid, for example, backprojection reconstruction in computed tomography (CT) or magnetic resonance imaging (MRI) and diffraction tomography.
In many real-world situations, the data is not uniformly sampled. In spiral MRI, for example, the non-uniform sampling is due to time varying magnetic gradients. Typically, allowing non-uniform sampling can significantly shorten MRI data acquisition time.
In MRI, data is acquired into a frequency domain space called k-space, which is the Fourier transform space of the image. An image is usually reconstructed from the k-space by applying an FFT to the data in the k-space. A major difference between MRI methods is how k-space is traversed to acquire the data. For example, in spiral MRI, data is acquired along a spiral trajectory in two-dimensional k-space, while in two-dimensional Fourier transform (2DFT) MRI, data is acquired along individual rows in k-space. In three-dimensional MRI, the k-space is three dimensional.
To allow signal and image processing techniques such as the FFT to be applied to non-uniformly spaced data points, the non-uniformly spaced data points can be interpolated onto a uniformly spaced grid, a process known as gridding. Gridding was originally introduced into medical imaging by O'Sullivan in “A Fast Sinc Function Gridding Algorithm for Fourier Inversion”, IEEE Trans. Med. Imaging, MI-4:200-207, 1985 and by Jackson et. al in “Selection of A Convolution Function for Fourier Inversion using Gridding”, in IEEE Trans. Med. Imaging, MI-10:473-478, 1991.
In the gridding algorithm presented by O'Sullivan, resampling (reconstruction) of data (in k-space) is applied as follows:
(a) Multiply the trajectory data by an appropriate density compensation function to compensate for the varying density of sampling in k-space.
(b) Convolve the density compensated trajectory data by an appropriate window function.
(c) Re-sample onto a uniformly spaced Cartesian grid by multiplying a comb function.
(d) Perform an FFT on the redistributed set of data points to get an image.
(e) Deconvolve the transformed data to remove apodization of the convolution kernel by dividing the image data by the transform of the window function.
For real time MRI imaging, e.g., dynamic imaging of the heart, an important requirement for the reconstruction process is that it has to be done fast enough to maintain the rate at which the data are acquired.
Various solutions have been applied to overcome the speed limitations of reconstruction processes. Some solutions have used special hardware or distributed computer systems. These solutions, however, are costly due to the special hardware required. Other solutions have minimized the number of computational steps allowed between sequential images. For example, a Kaiser-Bessel window may be used as the convolution kernel, rather than a sinc window, to reduce the computational complexity. However, for high speed applications such as real-time systems where images are acquired continuously, the Kaiser-Bessel convolution may itself be too computationally intensive.
Another important requirement for any reconstruction process is that it must provide an image of sufficient quality for use in medical diagnosis. For example, degradation of image quality may occur towards the edges of the generated image in the form of “cupping”. Cupping is where the intensity profile is lower (or higher) at the center of the image than at its ends and wings is where there is an overrun of the signal beyond the ends of the image carrying portion of the image. One solution to such degradation is to interpolate onto a 2N×2N grid, rather than onto an N×N grid (oversampling). The result is then post-compensated and only the central N×N portion of the post-compensated result is persevered. Most of the artifacts are outside this central portion. While this method works well to improve image quality, this technique increases the number of points for the FFT, by a factor of four, which considerably increases the complexity of the computation.
Thus, there remains a need for a reconstruction process that provides an image of sufficient quality for use in medical diagnosis, while being fast enough to maintain the rate at which the data are acquired and minimize latency.